Aircraft: LARGE TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Cruise
Anomaly: Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy

This event cycle began with a call I made to the dispatcher from my hotel room in LLBG just prior to getting on the crew bus for the drive to the airport and subsequent flight home. He mentioned that there was an existing MEL write up on our aircraft. This read; 'Left intermediate press. sensor inop. (Bleed Press Sensor L).' He said that the flight crew that had brought the aircraft had had a rather lengthy conference call on their flight with [Dispatch] and [Maintenance Control] about this write up. My dispatcher indicated (and I concur) that there was nothing for the flight crew to do about this write up; no switches to re-configure and no legal or practical limitations on the operation. Once in our own aircraft doing preflight and cockpit set up duties; we did not see any STATUS messages connected with this MEL. So we sent an ACARS inquiry to [Maintenance Control]. Was the system's ability to generate a status message disabled or locked out by maintenance action in concert with the MEL? Unfortunately; we never received a response back and I did not follow up.We then took off and climbed normally. Then early in cruise; at 32;000 ft after the relief crew had left the cockpit for their rest break; the first officer (FO) and I got an EICAS caution message 'BLEED OFF ENG L.' We read the checklist guidance which basically said 'The engine bleed valve is closed because of a system fault.' We also broke out the flight manual and looked at pneumatic schematic along with pulling up the aircraft 'AIR' synoptic. All the aircraft pneumatic system isolation valves had opened up properly and all systems were being supplied. Next; we SATCOMed [Dispatch] and got a patch with [Maintenance Control]. Our question was whether the EICAS' BLEED OFF ENG L' was at all related to the previously mentioned MEL. We were assured that it was not; that the two anomalies were to be considered separate in that the MEL problem did not cause the EICAS. We also discussed the fact that we still had engine anti-ice available for the L engine as well as a full up wing anti-ice system for both wings. We felt OK with our situation; [Maintenance Control] had nothing more to add and neither did [Dispatch]. When the relief crew took over the cockpit; I had failed to brief them on the EICAS 'BLEED OFF ENG L' that we had dealt with four hours previously. Therefore; they soon launched their own subsequent inquiry when we were over the Atlantic (ETOPS airspace) and were rightly concerned that we were operating with only one air source (the R engine bleed) to supply our pneumatic needs. A loss of that one source would have necessitated an eventual descent to 22;000 to be able to draw APU supplied pneumatic air. Not a great thought. Granted; this EICAS 'BLEED OFF ENG L' occurred after takeoff (after the point of dispatch). No violation; but in retrospect; how good an idea was this operation? I cannot think of a FAR that we violated but it seems that safety was eroded. In this day and age when we want to operate every flight successfully; I believe that it is important to self-reflect and ask 'Did I contribute to a 'Morton Thiokol' mentality where 'By golly; we are going to complete the mission; here boys!' We don't want that mindset to pervade our cockpits to where we develop a culture of 'mission completion' at all costs. Perhaps I succumbed to this mentality myself briefly. Yet; we all know as well that if we want a risk free world; we need to all stay in bed all day curled up in the fetal position. In that kind of world; airplanes do not fly and airlines do not exist. Measure; reasoned risk allows us to operate and make money (since we are not a utility). Yet; due to [a] recent aircraft fire; the very aircraft that we flew was being taken out of a planned 24 hr. service visit to be sent back out. At some point; we may have a situation where we end up fixing that aircraft in a different location as opposed to our own hangar. I'm sure decision makers in those cases; compared to the rest of us; have way more brain power and are making far more money. I rest my case. A flight crew experienced an EICAS message 'BLEED OFF ENG X' early in cruise. They complied with the checklist and consulted with the Dispatcher and Maintenance and continued the flight. The reporter wonders if it was a good idea to have continued the operation into ETOPS airspace.